A quantum search method for quadratic and multidimensional knapsack problems

TL;DR

This work extends the Quantum Tree Generator (QTG) to the 0-1 Quadratic Knapsack Problem (QKP) and the 0-1 Multidimensional Knapsack Problem (MDKP), building polynomial-depth quantum circuits that generate superpositions of all feasible solutions and using quantum maximum finding to extract high-quality solutions. By incorporating linear and quadratic profits and handling multi-dimensional capacities, the authors implement problem-specific state preparation for amplitude amplification, and benchmark performance against Gurobi under optimistic, fault-tolerant hardware assumptions with scalable runtimes up to thousands of variables. The results show potential quantum advantage for QKP, particularly when optimality gaps are moderate, but MDKP remains challenging, with limited or instance-dependent speedups. Overall, the study underscores both promise and current limits of quantum optimization for structured combinatorial problems, highlighting the need for further algorithmic refinement and hardware progress to achieve robust, generalizable quantum speedups.

Abstract

Solving combinatorial optimization problems is a promising application area for quantum algorithms in real-world scenarios. In this work, we extend the "Quantum Tree Generator" (QTG), previously proposed for the 0-1 Knapsack Problem, to the 0-1 Quadratic Knapsack Problem (QKP) and the Multidimensional Knapsack Problem (MDKP). The QTG constructs a superposition of all feasible solutions for a given instance and can therefore be utilized as a promising state preparation routine within amplitude amplification to produce high-quality solutions. Previously, QTG-based search was tested on the 0-1 Knapsack Problem, where it demonstrated the potential for practical quantum advantage, once quantum computers with a few hundred logical and fully connected qubits are available. Here, we evaluate the algorithm's performance on QKP and MDKP against the classical solver Gurobi. To facilitate large-scale evaluations, we employ an advanced benchmarking technique that enables runtime predictions for instances with up to 2000 variables for QKP and up to 1500 variables and 100 constraints for MDKP. Our results indicate that QTG-based search can produce high-quality solutions with competitive runtimes for QKP. However, its performance declines for MDKP, highlighting the challenges quantum algorithms face when tackling highly constrained optimization problems.

Figures (2)

• Figure 1: The circuit for QKP builds on the circuit QTG for KP Wilkening2023AQuantumAlgorithmForTheSolutionOfTheKnapsackProblem, and takes into account the two-control-one-target gates for adding the quadratic profits on the profit register.
• Figure 2: QTG-based search vs Gurobi: performance analysis. We benchmark the algorithms on relevant benchmark instances with up to $2000$ variables for the QKP and $1500$ variables and $100$ constraints for the MDKP. Each data point represents the time required for Gurobi to obtain an incumbent solution and for the QTG-based search to find an equally good or better solution under benevolent assumptions for the quantum algorithm, see \ref{['subsection:AssumptionsOnTheQuantumProcessingUnit']}. The markers' sizes and colors denote the instance's size and the solution's quality, respectively. Assuming a quantum cycle time limit of $1ns$Chew2022UltrafastEnergyExchangeBetweenTwoSingleRydbergAtomsOnANanosecondTimescale, we observe that for the given benchmark sets the QTG search has the potential to outperform the state-of-the-art solver Gurobi. This is especially noticeable when dealing with problems that are more difficult to solve for classical algorithms (QKP).